A Note on the Adjacent Vertex Distinguishing Total Chromatic Number of Some Cubic Graphs

Abstract

An adjacent vertex distinguishing total coloring of a graph \(G\) is a proper total coloring of \(G\) such that no two adjacent vertices are incident to the same set of colors. The minimum number of colors needed for such a coloring is denoted by \(\chi_{at}(G)\). In this note, we prove that \(\chi_{at}(G) = 5\) for some cubic graphs.